Half-Angle Formula Calculator

The half-angle identities express sin(θ/2), cos(θ/2), and tan(θ/2) in terms of cos(θ). They are derived from the double-angle formulas by replacing θ with θ/2:

Starting from cos(2x) = 1 − 2sin²(x), replace x with θ/2: cos(θ) = 1 − 2sin²(θ/2) → sin(θ/2) = ± √((1 − cos θ) / 2)

Starting from cos(2x) = 2cos²(x) − 1: → cos(θ/2) = ± √((1 + cos θ) / 2)

For tangent, a cleaner form avoids the ± ambiguity: tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ

The ± signs on sin and cos depend on the quadrant of θ/2, not θ. That is the step people skip. If θ = 200°, then θ/2 = 100°, which sits in the second quadrant, so sin(θ/2) is positive and cos(θ/2) is negative. Reading the quadrant off θ instead would have put you in the third and given both of them the wrong sign.

The calculator does two things side by side. It computes sin(θ/2) and cos(θ/2) directly, which is always right, and it also runs the identity with the sign chosen from the quadrant of θ/2, then reports how far apart the two answers are. They agree to about fifteen decimal places, which is the point: the identity is exact, and the ± is the only part that needs judgement.

Why two forms for tangent? Because each one breaks somewhere. sin θ / (1 + cos θ) divides by zero at θ = 180°, where cos θ = −1. The other form, (1 − cos θ) / sin θ, divides by zero at θ = 0° and 360°, where sin θ = 0. Between them they cover the whole circle, and the calculator prints whichever is usable along with a note when one is not.

When do half-angle formulas come up? Integration is the main case, specifically integrating sin²(x) or cos²(x). The power-reduction formulas are the same identities rearranged, and they turn those integrals from awkward to routine. They also give exact values: sin(22.5°) = sin(45°/2) = √((1 − √2/2)/2), which simplifies to √(2 − √2)/2 ≈ 0.38268.

There is a second reason to care, which is the Weierstrass substitution. Setting t = tan(θ/2) turns sin θ into 2t/(1+t²) and cos θ into (1−t²)/(1+t²), so any rational function of sine and cosine becomes a rational function of t, and rational functions can always be integrated. It is one of the few genuinely universal techniques in calculus, and it is built entirely on the tangent half-angle formula above.